Talks

This past weekend, we gave talks at Bates. They went well, because we had practiced them and we knew how to communicate effectively and all of that. We gave talks on Friday night, and then the conference continued all day Saturday, so people kept coming up to the four of us and telling us how great our talks were, and asking us about Williams, and telling us how great it was that four females were doing math and how we should all go to grad school. After all, we kind of stood out, four young females hanging around together at a math conference.


Professor Morgan and the 2005 Geometry Group

Now that we have a good background in what we are doing, the research will begin in earnest.

30 pages

I just finished writing and printing out my 30-page paper. Five pages of that were contents, appendix, bibliography, etc. But the other 25 were writing. And now I'm done, and I am very proud of myself. I wonder if my professor will read it. She will be receiving 50 papers, all at least 10 pages in length. But she is a professor of education, so she probably will. And it's 2:30 AM. That's late.

Williams-Mystic

So, the suspense is over: I was accepted to Williams-Mystic! Golf clap. I think the admission rates for this program are exceedingly high. Williams-Mystic is "the maritime studies program at Mystic Seaport" which is a Williams program wherein a group of students, several of whom are from Williams, go to Mystic, CT and take four classes about maritime things and learn to sail boats and whatnot. It is a semester program. I will be going on this program. This means that I might drop the psych major. Sorry about that, psych major. I'd choose a thesis over a psych major. In case you want to know more, here is the link: Williams-Mystic (but their site is not so excellent, sorry to say).

The Möbius strip

Today, I decided to do a little Möbius strip project that I have been thinking about for a while. While I was doing this project, I realized that I didn't understand exactly how the Möbius strip works, so I decided to do a little experiment.

("What are you doing?" asked my S.O. "I'm doing a math experiment," I said. "There are no math experiments," said my S.O., "math is not an experimental science." "Then I'm performing a miracle," I said.)

I took a strip of paper and taped it together to make a Möbius strip, with one half-twist. Then along the left side of the strip, I made a purple line, and next to it on the right side of the strip, I made a yellow line. Note that this means that along the whole strip, there were adjacent purple and yellow lines, and left and right mean nothing; they're just to orient you.

I was wondering what would happen to the lines when I cut the strip the long way down the middle, because I figured that it couldn't possibly just end up with the yellow line switching to purple or something; it had to be that the yellow line would be on one side and the purple line would be on the other. And that is what happened! Wow.

So I figured the next thing should be to see what happens when you add more stripes. So to the right of the purple line I drew a blue line, and to the left of the yellow line I drew a red line, both all the way along the whole strip. So now on one side of my strip there were parallel blue and purple lines, and on the other strip there were parallel yellow and red lines.

("Logical fallacy," you say: "A Möbius strip has only one side." Correct, but once you cut it in half the long way, it ends up having two sides, and it's not a Möbius strip anymore. Pity, I know, but it's the truth. Instead of one half-turn, a cut-in-half Möbius strip has two full turns (four half-turns), I think.)

The idea of doing this was that you could not end up with red, yellow, purple, and blue each on their own side, because there are only two sides. So something drastic had to occur. This is probably common knowledge, but here is what happens when you cut an already-cut Möbius strip in half again: You get two interlocking twisted strips! They each have four half-turns. And they're not just singly linked, like the Olympic Rings, but are kind of doubly linked.

Naturally I tried cutting them in half again, and I got four rings which are all interconnected in a kind of knotted-up way. That wasn't very interesting, but I knew you'd be wondering. I should try it with a bigger piece of paper next time. For a more interesting experiment, try this, suggested by Mathworld (link below): "In addition, two strips on top of each other, each with a half-twist, give a single strip with four twists when disentangled." It's true! It happened to me! Whoa.

For anyone who is very interested in the Möbius strip (and you had better be interested, Ms. 3.95) check out the animated Möbius strip on Mathworld. You have to have some sort of Java to see it, I think, but if you do, you can click and drag to spin it around, and if you kind of drag and let it go as though you are throwing it, it will spin in the direction in which you threw it, so long as you keep your cursor within the boundary of the graphic. Awesome. I think that I was talking about this awesome graphic when someone gave me the idea for the project that inspired me to investigate the particulars of the Möbius strip -- a project that I will soon be sending to you, Ms. 3.95, which is why you had better be interested, because today, I decided to do a little Möbius strip project that I have been... yes, you see, it all comes together at the ends, just like a Möbius strip.

05.05.05

That's what today is -- 05/05/05, or as I like to call it, 05.05.05. If you were awake at 05:05 this morning, that would be really special -- 05:05:05 05.05.05. But if you missed it this year, there's always 06:06:06 06.06.06 next year, and every year until 2012 (12:12:12 12.12.12). Professor Morgan pointed this out in -- you guessed it -- Math 305. Too bad the class meets at 9:55, although that has its fair share of 5's.

Today Vojislav and I were working in the math library on topology, and Professor Morgan was talking with us as he was about to leave. Then it apparently occurred to him that Vojislav is from last year's Geometry Group, I am going to be in this year's Geometry Group, and he was going to an open house hosted by a member of the 1990 Geometry Group (the one that published the paper proving that the double bubble is perimeter-minimizing in R² in Euclidean space). So he invited us to come along. So half an hour later, Vojislav and I were on our way to Pownal, VT where we got to see this guy's new house. It was a nice house, with a big field going down to a marsh. And Professor Morgan took us to get ice cream afterwards. I had graham cracker flavored ice cream (on his recommendation). It was yummy.

We went out to look at their field, which slopes down towards the marsh, and I found myself trying to figure out what had happened in the past to make it look like this, just as we did at the Mountain School. I determined that the part by the house had been a field for a while, because it was flat and mown. There were about six trees dispersed through this field, and since they all looked different, I decided that they were different species of trees that had been planted, rather than growing naturally from seeds. On the edge of the field at the bottom there was a peculiar swath leading down to the marsh with no trees on it, which really perplexed me. To the left was a big tree on the corner, which means that it was on the edge of the field, and behind the big tree to the left were trees about 10 inches in radius, which were all about the same size. By contrast, on the right of the cleared swath was an area full of little trees, pretty tall (20-30 feet) but all under 4 inches in diameter. So I figured that area had been pasture much longer than the part on the left, and had at some point been left to grow into trees, at which time all the little trees grew in.

For anyone who kept reading, try this, which is our Geometry Group homework:

Given that a circle is perimeter minimizing for given area in the Euclidean plane (i.e., R²), prove that a semicircle is perimeter minimizing in the Euclidean halfplane (i.e., just the part above the x-axis), and that a quarter circle is perimeter minimizing in the first quadrant.

I have proven it. It is a very short proof.
If you finish that, try this:

Gaussian space is endowed with variable density, so that points near the origin are denser than points far away from the origin. The distribution follows a normal distribution (bell-shaped curve). So area is worth more near the origin, but perimeter also costs more near the origin. What is the perimeter-minimizing shape to enclose a given percentage (say, 10%, or 30%, etc.) of the Gaussian plane?

Hint: it's a shape you've heard of.
Next hint: It's not a circle. But that was a good guess.

HRUMCXII

On Saturday we hosted the HRUMC XII at Williams. For those of you who don't know, that is the Hudson River Undergraduate Mathematics Conference, year 12. Ken Ribet came to speak. Do you know who he is? Probably not. You will soon enough.

A long time ago, a guy called Fermat said that there are no nonzero integer solutions to the equation aⁿ + bⁿ = cⁿ for n > 2. He said he had a clever proof that was just too big to fit in the margin he was writing in, and so mathematicians tried for many years to prove what was known as "Fermat's Last Theorem." Then in the middle of the 20^th century, over in Japan two mathematicians called Taniyama and Shimura were doing some seemingly unrelated mathematics, leading to the conjecture that modular forms and elliptic curves were essentially the same thing. Then another mathematician gave a plausibility argument that if Fermat's last theorem were false, it would create a bizarre function that would not fit into the structure of the so-called Taniyama-Shimura conjecture, rendering Fermat and Taniyama-Shimura logically equivalent -- but he couldn't prove it. So Ken Ribet came along and proved it. And then Andrew Wiles proved the Taniyama-Shimura conjecture, which proved Fermat's Last Theorem, which is what all the fuss was about with Fermat's Last Theorem finally being proven.

Here are some pictures of us Williams people with Ken Ribet:


Professor Pacelli, Professor Ribet, Neil, me, and Brian.


Ken Ribet and me.

I talked to the people who were giving a talk on "A Problem Oriented Approach to Geometry." Here was their abstract:

We think it may be possible to organize geometry courses around a set of problems. This approach seems unusual. We have not collected enough problems for a complete course but we have started such a collection. We will share examples of problems we think are interesting, challenging and instructive.

So before the conference I e-mailed them and told them that I had learned not only geometry, but also everything else after algebra and all the way up through BC calculus via a problem-oriented approach. I gave them the URL for the Exeter teaching materials and they wrote back and said something like "that's very interesting, thanks, I'll check it out sometime." They did not sound very interested. But when I went to talk to them on Saturday, they were very excited about the curriculum, said it was the best thing ever (an exaggerated paraphrase) and said that they were going to print out the whole thing (no joke, all 400 pdf pages). I told them it was most definitely the best math curriculum ever because people actually learn math, and they don't forget it, even over the summer, even now that I haven't done that math for two years I still remember the problems from ninth grade and the strategies to solve them. It is the best math curriculum ever. Don't waste your time with a textbook. Go straight to the heart of real mathematics and actually learn something. Please.

I also gave my talk on Latin squares. People actually came to watch it. It was astounding. People actually came into the room right before my talk started, to watch it. It was amazing. Who would've though that people actually would come? And the guy chairing the session memorized the number in my talk. That was nice of him, but I felt kind of bad because it was a really long and somewhat useless number. It was actually a pretty engaging and informative talk, if I do say so myself, though not as enlightening as talks that were of a higher level. And 500 people came to the conference. That was a lot of people. And it rained. And Kate Kraft was there. But she didn't come to my talk.

Oh, and the number in the title is the total number of 10x10 Latin squares. I assume you were wondering.

I got into Williams-Mystic! Yay! Take that, Williams-Exeter Programme in Oxford! Take your humanities majors and your stellar GPAs and leave all the engaged mathematical learners back in this country; I'm going to Mystic! (Williams-Mystic site)

ALICE

This week, my cognitive science lab was to "chat" with a robot that won the Turing Test competition. It doesn't quite pass the Turing Test -- the first robot to do it wins a $100,000 prize -- but the closest one each year gets $2,000 anyway. Check out the conversation I had with ALICE (link provided because it is an amazingly good robot):

ALICE: Are you using Netscape or Explorer?
Human: No.
ALICE: Are you serious?
Human: Yes.
ALICE: I can't believe it.
Human: I'm using Firefox.
ALICE: How long have you been using it?
Human: A couple of months.
ALICE: I did not know that.
Human: I think everyone should switch over.
ALICE: Since when? That's a rather sweeping generalization.
Human: Since Explorer started putting spyware on my computer.
ALICE: Interesting explanation. It's too bad we have to use a monopoly like microsoft.

That honestly sounds like a real person talking, does it not? Unfortunately, the robot is not always perfect; these are the next few lines of our "conversation":

Human: That's the thing, we don't, we can use Firefox.
ALICE: Yes it is one of the thing we do not we can use Firefox.

and another time:

Human: Just believe that you can, and anything is possible.
ALICE: And possible is believe that me can and anything.

In case you're interested, here are a few other winners of the Turing Test competition: 2003, 2002, 1999 (ALICE won in 2000 and 2001).

I'm thinking about your Odd Quad question. I'll get back with a response eventually.

Housing Pick Number One

Pick #1!
Pick #1!
I can live wherever I want! Yaaaaaaaaay!
Take that!

The coolest theorem

Today we learned The Brouwer fixed-point theorem for the disc:

If f:B² --> B² is continuous, then there exists a point x in B² such that f(x) = x.

This says that if you have a circle (the inside too, not just the outside edge), and you take the points on that circle and move them around in a continuous way, you will always have a point that doesn't move.

More impressively, take two identical sheets of paper and lay them on top of each other. Crumple one up and set it on top of the other one. There is some point that is directly above the point where it started. You can flip the crumpled paper over, move it around -- there will always be a point that doesn't move.

"But," you say, "I thought you said it was true for a disc! A piece of paper is not a disc." Silly, you're thinking that circles and rectangles are different! But they're topologically equivalent, just like you and a pencil eraser (unless you have pierced ears).

HTML tutorial

The Internet boasts a wide variety of HTML tutorials, but none are quite good enough. So I wrote my own. I believe that this guide, unlike all the others, concisely and effectively guides the reader from knowing nothing about HTML to having a complete Web page with just about everything anyone would want to have on it. This tutorial grew out of the HTML class I taught through Free U over Winter study.

Here it is.
And here is a mirror for all the times (such as right now) when WSO student Web pages are down.

Enjoy.

I got a 10 on the Putnam.

Yup. That's what I did.

I also got 6, 6, 7, 4.5 hours of sleep the last four nights, so now I'm going to go make sure all of that was marginally worth it by editing the papers I was writing all that time. I GOT A 10 ON THE PUTNAM. DO YOU HEAR ME? I AM A MATH MAJOR! So WHY, oh WHY do I have to write so very very many papers?

P.S. The score on the Putnam is out of 120. Then again, the median is 1.

56 miles

Last week, I ran 56 miles (see running [b]log). And I did this while taking the requisite two rest days out of seven, which means that I averaged -- AVERAGED -- 11 miles a day. To paraphrase a person called (d)avid, who was not talking about anything even remotely resembling running:

Running 11 miles in a day means you got lost -- Averaging 11 miles a day means you're just plain crazy.

The indiscrete topology

In Topology class on Monday, we were discussing whether certain sets were compact in certain topologies. Now, I'm okay with the product topology, the uniform topology, and the box topology. I'm okay with the discrete topology and the finite complement topology. But then someone mentions the indiscrete topology. So I'm wondering, just what is this indiscrete topology? I could not remember what it was. Reasoning: Maybe it's the opposite of the discrete topology. But no, the finite complement topology is kind of the opposite of the discrete topology. But I didn't want to ask, because everyone else seemed to know. After class, I looked it up. Do you know what the indiscrete topology is? No, you don't. It's just the empty set and the whole space. Is that absurd or what? I know -- topologies aren't allowed to have two names! The other name, and the more logical name, for that topology, is "the trivial topology." Because honestly, that is the most trivial topology you could ever think of. Honestly. You can write it in two symbols! It is the only topology that can make that not-so-lofty claim. Trivial.

So I'm going to come up with other names for the other topologies too. For instance, if I don't know the answer to a problem such as "which topology has such-and-such property?" on a test, I'll say, "the inconsiderate topology." Then if anyone says that is wrong, that the real answer was the box topology, I will simply say, "oh, 'the inconsiderate topology' is just my other name for the box topology, like the indiscrete topology, except that this one goes so far as to be inconsiderate." Because if the trivial topology can get a special name, I think every topology deserves a special name.

Latin squares: 9982437658213039871725064756920320000 permutations inside the box

A Latin square is an nxn grid of symbols in which each of n symbols appears once in each row and each column. We will discuss orthogonality, Cayley tables, and the analogy of Latin squares to the rook problem, and you will discover how Latin squares apply to tire rotation. What does the number in the title have to do with Latin squares? Come find out.

***

Yup. That's my title and abstract for my upcoming talk at the Hudson River Undergraduate Math Conference, which will be held at Williams College on April 30. Sounds like fun. You should stop by; my talk is only 15 minutes long.

Different stimuli

QOTD:

We'll think about you and 5th dimensional and beyond bubbles while gazing out at the water and sailing off shore this summer. It's good that we all
get our highs from different stimuli. Go for it!

More on the context later.

Anchors Away

I have become the Webmaster for Anchors Away, the Williams anti-anchor-housing group. I put up a Web site for it yesterday evening, and it's already gotten 200 hits -- including hits from Indonesia, Japan, and Finland, though the vast majority is from Williams. See here:

http://wso.williams.edu/orgs/anchors-away

I am adding names to the list of opposed students as soon as they land in my inbox, and I am posting news within 15 minutes of hearing of it, so I hope that this will be a good resource. So go visit.

It looks pretty much like every other Diana Davis site, except that this time, it actually matters, kind of.

Free U HTML course: Final products

Over Winter Study, I taught a course on how to make a Web page. Now the course is over, and several of my students actually stuck it out to the end and made actual, functional Web pages. Here they are:

Jerry He '08
Delwar Hossain
Maksym Kepskyy
Lily Li '08
Irina Zhecheva '08

Some of these clearly show signs of the entertaining examples I used in class (the bright backgrounds, the cow...). But I think that they are really nice sites, and I hope that these students will continue to update them in the future.

Maksym (who spells his name many different ways) plans to make his site into an extreme sports page, so that people will come and read about different sports, and watch 10-second videos of them, and read short statements from people who do the sports about why they like them. He hopes that people will then decide to try out these sports on their own. He already has a sound introduction (in his native language) and a video on the site, which are very worth checking out.

Did anyone notice that no one called Smith or Johnson stuck it out to the end? Hmmmm. I had 19 students (over half were CDE students) for the first class. They all created nice pages, and then only five came back the next week to learn how to put them on the Internet. Hmmmmmmmm.

Beware of the Snow Plow.

This morning, I got hit by a snow plow.

It was not a very big snow plow -- it was one of those little orange ones they use to clear the sidewalks -- but it hit me nonetheless.

When I left my dorm, I found myself upon a two-inch layer of snow that no step had trodden black. This was fine, but when the snow plow came along, I figured I'd walk along behind it so that I would be walking on something plowed. I was walking rather close to the plow, because it was driving slowly and I was late.

Then all of a sudden, as I was walking behind it, the snowplow went into full-speed reverse! I shrieked and grabbed onto the back of the vehicle and ran backwards to avoid getting driven over, leapt into the snowbank as soon as I could, and ran away through the deep snow in the middle of the quad. The snowplow operator called after me, but I shouted that I was fine and kept running.

I was not very much hurt at all; the back of the snowplow slammed into my knees, but otherwise there was no lasting damage. However, when I took the container of yogurt out of my jacket pocket, I found that it was quite smashed.

Let this be a lesson: NEVER follow a vehicle that doesn't know you're there, even if it doesn't have a "do not follow" sign on the back of it. I was lucky.

Super clever math

First, it snowed. Then it rained and froze again and such, so there was kind of icy snow. Then it snowed about 3". Then I went sledding. The first few runs were great -- really fast, and smooth, and all -- but then the snow got "sledded off" so there was just this icy snow stuff. That was perfectly fine for sledding, so I continued. When I looked at my tray afterwards, though, there are all these deep lines in it from the sharp edges I sledded over. Oops! Last year there was so much snow that this was never an issue.

I made the quiz that the regular math students took today, and I made a worksheet that the advanced math students are doing for homework tonight. I was really proud of that worksheet. If they can exert enough effort and brain power to do it, they will learn something without having to explicitly have it taught to them, just like the Exeter math curriculcum. It goes like this:

(5) 6 people are sharing 9 pizzas. (Don’t worry, they’re mini pizzas.)

(a) If each pizza is cut into four slices and everyone eats the same amount, how many slices does each person eat? (6) (slices/person)
(b) If each pizza costs $5 and everyone pays an equal amount, how much does each person pay? ($7.50) (dollars/person)
(c) If each pizza costs $5 and is cut into 4 slices, how much is it per slice? $1.25 (dollars/slice)
(d) Multiply your answers to (a) and (c). Why is this the same as your answer for (b)? (6 x $1.25 = $7.50)
(e) If you didn’t already, add units to your answers. For example, the units of (a) are slices/person.
(f) Redo (d), this time including the units when you multiply. (6 slices/person x 1.25 dollars/slice = $7.50 dollars/person

The answers to each problem are in the parantheses, but of course the students don't have that on their worksheets. The idea is that if they can get through (d), they will make a synapse connection. If they can do and understand (f), they will figure out dimensional analysis (the thing you use to convert meters/sec to miles/hr) before I teach it to them tomorrow. So tonight I have to figure out exactly what I am going to do in class tomorrow. Sounds like fun!

My HTML class

Last night I had the first session of my Free U HTML class. I had 30 people signed up for the course, so I actually had to move it to the computer lab in Jesup, and it was good I did -- 19 people showed up! Over half of these were graduate economics students from the graduate department. (Yes, Williams has a graduate department -- to be a student there, you have to be a practicing economist in a developing country, so needless to say, I had some pretty cool students.) So far as I know, one of my students was actually from Afghanistan, and I think another is Bangladeshi. I wanted to take attendance just so that I could ask them where they were from, but I chickened out...

In the two hours, we actually got a lot done. In other words, I told them absolutely everything I know about HTML, and it only took me two hours. They ended up with kind of funny pages, which were generally bright green, had pictures of stretched-out purple cows, and said things like "This is a cow" and "Williams is a really good place." This was because my example page, which I used to explain each thing that I taught, was bright green, had pictures of stretched-out cows, and said things like that. You can see it here because I actually showed them an example of uploading by uploading my own page. I think I obliterated an early version of my Web page by doing that (it was originally hosted at that address) but oh well, these are the side effects of effective teaching.

Next week I plan to have them work independently writing a real Web page, and for our final meeting they will all hopefully have WSO accounts, so I can teach them how to upload to the Internet and update their pages later. Since most of them are in the graduate econ program, they have real motivations for wanting to learn to make a Web page, so I am confident that they will think of good pages to create. My plan is to walk around and answer questions, and look at their code and find the errors when things come out looking wrong. I was so happy about this last night, because I was actually able to find errors! Things like < img scr =... It kind of makes up for my feelings of extreme ineptitude when I was taking CSCI 134 last spring. Kind of.

A restful day

Today, I got up at 10:30. That wouldn't seem so surprising, since it's a weekend and I'm a college student, but I also went to bed at 10:00. So I slept for over 12 hours. Wow. And the crazy thing is, I wasn't sleep-deprived; I slept for nine hours the previous few nights. The idea of going to bed at 10:00 on a Saturday night is not to be a social reject; I'm trying to move my bedtime earlier so that I will be able to fall asleep when I go to bed at 9:00. This way, when my alarm goes off at 6:30 in the mornings for school, I will be well-rested and not yawn all day.

Today when I was running, I happened to look up at the sky just when a cloud had turned into a rainbow! It was a most amazing sight, one that I have never seen before: the cloud was multicolored! As the wind blew, it spread out the cloud, so the colors didn't last over 30 seconds. I managed to take a picture of it, and the nice people who wrote back to me on my WSO blog post about it have informed me that what I saw was a sun dog. Neat!

I spent the rest of the day in my room, typing up an explanation of the basics of HTML for the Free University class I am teaching. I managed to give a good introduction to pretty much everything anyone would want to know, and I kept it relatively short (eight pages). When I was finished, I realized I forgot to put in anything about page anchors! Oh no! I doubt anyone will ask... I did put in just about everything I know, and certainly everything I use on a regular basis, though. I think it's good to know HTML, because everyone uses the Internet so regularly that it is nice to know how it all works, from the inside.

Williams in the Snow

This morning, after going on a delightful run in the puffy snow, I decided to photograph it. I took some lovely pictures, which you should look at. You can see the ones that relate more to me here and the Williams-based ones on my Ephblog post.

Yes, run in the snow if you possibly can! It's lovely.